\(\mathrm{h(x) = (13 - 2x)(x + 5)}\)The function h is defined by the given equation. For what value of x...

1. TRANSLATE the problem information

• Given: \(\mathrm{h(x) = (13 - 2x)(x + 5)}\)
• Find: the x-value where \(\mathrm{h(x)}\) reaches its maximum

2. INFER the problem type and approach

• This is a quadratic function in factored form
• Since we need the maximum value, we need to find the vertex of the parabola
• Two efficient approaches: expand and use vertex formula, or use the midpoint of roots

3. INFER parabola direction

• When expanded: \(\mathrm{h(x) = -2x^2 + 3x + 65}\)
• The coefficient of \(\mathrm{x^2}\) is -2 (negative), so the parabola opens downward
• This confirms there is indeed a maximum point at the vertex

4. SIMPLIFY using the midpoint of roots method

• Set \(\mathrm{h(x) = 0}\): \(\mathrm{(13 - 2x)(x + 5) = 0}\)
• First root: \(\mathrm{13 - 2x = 0}\) → \(\mathrm{2x = 13}\) → \(\mathrm{x = \frac{13}{2}}\)
• Second root: \(\mathrm{x + 5 = 0}\) → \(\mathrm{x = -5}\)
• Vertex x-coordinate = \(\mathrm{\frac{\frac{13}{2} + (-5)}{2}}\) = \(\mathrm{\frac{\frac{13}{2} - \frac{10}{2}}{2}}\) = \(\mathrm{\frac{\frac{3}{2}}{2}}\) = \(\mathrm{\frac{3}{4}}\)

Answer: B. 3/4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that finding the maximum of a quadratic requires locating the vertex. Instead, they might try to substitute the answer choices or attempt to solve \(\mathrm{h(x) = 0}\) thinking that's what "maximum" means.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need the vertex and use the midpoint of roots approach, but make arithmetic errors when calculating \(\mathrm{\frac{\frac{13}{2} + (-5)}{2}}\). Common mistakes include:

• Forgetting to convert -5 to \(\mathrm{-\frac{10}{2}}\) before adding
• Dividing by 2 only once instead of twice in the final step
• Sign errors when working with negative numbers

This may lead them to select Choice C (3/2) or Choice D (23/4).

The Bottom Line:

This problem tests whether students can connect the concept of "maximum value" to "vertex of parabola" and then execute the vertex-finding process accurately. The key insight is recognizing this as a vertex problem, not a roots problem.